Theorem tendoplcbv 37913

Description: Define sum operation for trace-preserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013.)

Hypothesis

Ref	Expression
tendoplcbv.p	⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))

Assertion

Ref	Expression
tendoplcbv	⊢ 𝑃 = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))

Distinct variable groups:   𝑡,𝑠,𝑢,𝑣,𝐸   𝑓,𝑔,𝑠,𝑡,𝑢,𝑣,𝑇

Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑓,𝑔,𝑠)   𝐸(𝑓,𝑔)

Proof of Theorem tendoplcbv

Step	Hyp	Ref	Expression
1		tendoplcbv.p	. 2 ⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
2		fveq1 6671	. . . . 5 ⊢ (𝑠 = 𝑢 → (𝑠‘𝑓) = (𝑢‘𝑓))
3	2	coeq1d 5734	. . . 4 ⊢ (𝑠 = 𝑢 → ((𝑠‘𝑓) ∘ (𝑡‘𝑓)) = ((𝑢‘𝑓) ∘ (𝑡‘𝑓)))
4	3	mpteq2dv 5164	. . 3 ⊢ (𝑠 = 𝑢 → (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))) = (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑡‘𝑓))))
5		fveq1 6671	. . . . . 6 ⊢ (𝑡 = 𝑣 → (𝑡‘𝑓) = (𝑣‘𝑓))
6	5	coeq2d 5735	. . . . 5 ⊢ (𝑡 = 𝑣 → ((𝑢‘𝑓) ∘ (𝑡‘𝑓)) = ((𝑢‘𝑓) ∘ (𝑣‘𝑓)))
7	6	mpteq2dv 5164	. . . 4 ⊢ (𝑡 = 𝑣 → (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑡‘𝑓))) = (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑣‘𝑓))))
8		fveq2 6672	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑢‘𝑓) = (𝑢‘𝑔))
9		fveq2 6672	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑣‘𝑓) = (𝑣‘𝑔))
10	8, 9	coeq12d 5737	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑢‘𝑓) ∘ (𝑣‘𝑓)) = ((𝑢‘𝑔) ∘ (𝑣‘𝑔)))
11	10	cbvmptv 5171	. . . 4 ⊢ (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑣‘𝑓))) = (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔)))
12	7, 11	syl6eq 2874	. . 3 ⊢ (𝑡 = 𝑣 → (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑡‘𝑓))) = (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))
13	4, 12	cbvmpov 7251	. 2 ⊢ (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))
14	1, 13	eqtri 2846	1 ⊢ 𝑃 = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))

Syntax hints:   = wceq 1537   ↦ cmpt 5148   ∘ ccom 5561  ‘cfv 6357   ∈ cmpo 7160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-co 5566  df-iota 6316  df-fv 6365  df-oprab 7162  df-mpo 7163

This theorem is referenced by:  tendopl  37914